Monotone versus non‐monotone projective operators

For a class of operators Γ$\Gamma$, let |Γ|$|\Gamma |$ denote the closure ordinal of Γ$\Gamma$‐inductive definitions. We give upper bounds on the values of |Σ2n+11,mon|$|\Sigma ^{1,mon}_{2n+1}|$ and |Π2n+21,mon|$|\Pi ^{1,mon}_{2n+2}|$ under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that |Πn1,mon|